We investigate the statistics of selected rare events in a $(1+1)$-dimensional (classical) stochastic growth model which describes the evolution of (quantum) random unitary circuits. In such classical formulation, particles are created and/or annihilated at each step of the evolution process, according to rules which generally favor a growing cluster size. We apply a large-deviation approach based on biased Monte Carlo simulations, with suitable adaptations, to evaluate (a) the probability of ending up with a single particle at a specified final time $t_f$ and (b) the probability of having particles outside the light cone, defined by a “butterfly velocity” $v_B$, at $t_f$. Morphological features of single-particle final configurations are discussed, in connection with whether the location of such particle is inside or outside the light cone; we find that joint occurrence of both events of types (a) and (b) drives significant changes to such features, signaling a second-order phase transition.

中文翻译：

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描述随机幺正电路演化的经典随机模型中的罕见事件特性

我们调查了选定的罕见事件的统计数据 $(1+1)$维（经典）随机增长模型，描述（量子）随机单一电路的演化。在这样的经典公式中，粒子在进化过程的每一步都被创建和/或消灭，根据通常有利于增加簇大小的规则。我们应用基于偏置蒙特卡罗模拟的大偏差方法，并进行适当的调整，以评估 (a) 在指定的最终时间以单个粒子结束的概率${吨}_F$ (b) 在光锥之外有粒子的概率，由“蝴蝶速度”定义 $v_{乙}$， 在 ${吨}_F$. 讨论了单粒子最终构型的形态特征，结合这种粒子的位置是在光锥内部还是外部；我们发现 (a) 和 (b) 类型事件的共同发生会导致这些特征发生显着变化，标志着二阶相变。